If Axiom 3 (A3) fails, which other axiom will fail automatically?

In the context of the axioms of a vector space, Axiom 3 typically refers to the associative property of vector addition. This property states that for any vectors ( u, v, ) and ( w ) in a vector space, the equation ( (u + v) + w = u + (v + w) ) must hold.

If Axiom 3 fails, it means that there exists at least some set of vectors for which the associative property does not hold. The failure of this property can disrupt the structure of the vector space because it directly influences how we combine vectors.

Axiom 4 generally expresses the existence of an additive identity, stating that there exists a vector ( 0 ) such that for every vector ( v ), ( v + 0 = v ). The failure of the associative property could lead to situations where constructs involving the identity do not behave as expected when combining with other vectors. Therefore, if the associative property is not upheld, it can create inconsistencies in the way operations on vectors are interpreted, leading to the failure of the additive identity requirement (Axiom 4).

Thus, if Axiom 3 fails, Axiom 4 is likely to fail automatically as